1 / 17

Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium

Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium. ACO and ILS for the Quadratic Assignment Problem Christian Blum IRIDIA, Brussels, Belgium. Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium.

terry
Download Presentation

Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO and ILS for the Quadratic Assignment Problem Christian Blum IRIDIA, Brussels, Belgium

  2. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium The Qadratic Assignment problem • Given: • n facilities, n locations • A nxn matrix D keeping the distances between locations • A nxn matrix F keeping the flows between facilities • Goal: Find a permutation  on the n facilities s.t. is minimal

  3. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Pheromone representation Pheromone matrix: facilities 11 1n locations i1 in n1 nn for every location-facility pair (i,j) there is a pheromone value ij

  4. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Ant construction phase • As long as there are free locations: • choose a free location i at random • assign a free facility j by using the following transition rule:

  5. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework • Online delayed pheromone update rule for Ant System: • where Disadvantage: upper limits of pheromone values dependent on optimal function value

  6. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework Normalization of the online delayed pheromone update rule:

  7. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework Solutions can be seen as binary vectors of size n2 0 0 1 3 1 2 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 Also the pheromone matrix can be written in vector form in the same way: 11 12 13 21 22 23 11 12 13 21 22 23 31 32 33 31 32 33

  8. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework  The pheromone matrix as a vector is moving in the n2-dimensional hyper-cube Advantages: Pheromone values are limited from above by 1 Scaling of objective function values

  9. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Pheromone update MMAS in the hyper-cube framework:max=0.99 min=0.01 where where Setting dependent on branching factor

  10. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Restart mechanism • After convergence: Restart by resetting the pheromone matrix • Usually: = • Diversification (by global frequency): • gf(i,j) = # of solutions containing assignment (i,j) • if gf(i,j) high we set ij < c • if gf(i,j) low we set ij > c c c c c

  11. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Restart mechanism • Intensification (by global desirability): • inverse of the best value of a solution containing assignment (i,j) • if gd(i,j) high we set ij > c • if gd(i,j) low we set ij < c • Daemon actions: Local Search and short runs of Tabu Search dependent on the distance dominance value • (both LS and TS provided by the Darmstadt node)

  12. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP

  13. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Perturbation mechanism • Perturbation mechanism: • Choose a number of k assignments from the current solution • Randomly reassign the k facilities to the k locations avoiding the assignments in the current solution • Parameter:size (strength) of the perturbation k • 5 <= k <= n/2

  14. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Evolution of perturbation size • if new local minimum accepted: • k = k – 3 • if new local minimum improves best solution so far: • k = 5 (minimum for k) • if new local minimum not accepted: • k = k + 1 • if beginning of a (re)start: • k = n/2 (maximum for k)

  15. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Acceptance criterion  : current solution ‘ : perturbed solution after local search

  16. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Evolution of parameter T • if the last 3 perturbed solutions in a row got accepted: • T = T * 0.9 • if 5 solutions in a row didn‘t get accepted: • T = T * 1.1

  17. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Restarts and usage of local search Restarts if no improvements after a number of iterations Local Search and short runs of Tabu Search dependent on the distance dominance value of the problem instance (both LS and TS provided by the Darmstadt node)

More Related